The Fibonacci zeta function is the sum of the reciprocals
of the k-th powers of the Fibonacci numbers. It is conjectured
that all of these values are transcendental numbers for k=1, 2, ...
However, this known only for even values of k. Surprisingly,
the problem takes us into the world of modular forms, elliptic
functions and modern transcendence theory. In this talk,
we give a short survey of this fascinating chapter in number
theory.